1 Pertemuan 11 Peubah Acak Normal Matakuliah: I0134-Metode Statistika Tahun: 2007

2 Outline Materi: Peluang sebaran normal

3 Basic Business Statistics (9 th Edition) The Normal Distribution and Other Continuous Distributions

4 Peluang sebaran normal The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution

5 Continuous Probability Distributions Continuous Random Variable –Values from interval of numbers –Absence of gaps Continuous Probability Distribution –Distribution of continuous random variable Most Important Continuous Probability Distribution –The normal distribution

6 The Normal Distribution “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33  Random Variable Has Infinite Range Mean Median Mode X f(X) 

7 The Mathematical Model

8 Many Normal Distributions Varying the Parameters  and , We Obtain Different Normal Distributions There are an Infinite Number of Normal Distributions

9 The Standardized Normal Distribution When X is normally distributed with a mean and a standard deviation, follows a standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. X f(X)f(X) f(Z)f(Z)

10 Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

11 Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!

12 Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Only One Table is Needed Z = 0.12

13 Standardizing Example Normal Distribution Standardized Normal Distribution

14 Example Normal Distribution Standardized Normal Distribution

15 Z Cumulative Standardized Normal Distribution Table (Portion) Z = 0.21 Example (continued)

16 Z Cumulative Standardized Normal Distribution Table (Portion) Z = Example (continued)

17 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet

18 Example : Normal Distribution Standardized Normal Distribution

19 Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Z = 0.30

Finding Z Values for Known Probabilities Z Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = ?.6217

21 Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

22 More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam?

23 What percentage of students scored between 65 and 89? More Examples of Normal Distribution Using PHStat (continued)

24 Only 5% of the students taking the test scored higher than what grade? ? =86.16 (continued) More Examples of Normal Distribution Using PHStat

25 Assessing Normality Not All Continuous Random Variables are Normally Distributed It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a Normal Distribution

26 Assessing Normality Construct Charts –For small- or moderate-sized data sets, do the stem-and-leaf display and box-and-whisker plot look symmetric? –For large data sets, does the histogram or polygon appear bell- shaped? Compute Descriptive Summary Measures –Do the mean, median and mode have similar values? –Is the interquartile range approximately 1.33  ? –Is the range approximately 6  ? (continued)

27 Assessing Normality Observe the Distribution of the Data Set –Do approximately 2/3 of the observations lie between mean 1 standard deviation? –Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? –Do approximately 19/20 of the observations lie between mean 2 standard deviations? Evaluate Normal Probability Plot –Do the points lie on or close to a straight line with positive slope? (continued)

28 Assessing Normality Normal Probability Plot –Arrange Data into Ordered Array –Find Corresponding Standardized Normal Quantile Values –Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis –Evaluate the Plot for Evidence of Linearity (continued)

29 Assessing Normality Normal Probability Plot for Normal Distribution Look for Straight Line! Z X (continued)

30 Normal Probability Plot Left-SkewedRight-Skewed RectangularU-Shaped Z X Z X Z X Z X